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Isaac Goldbring

Publications and source records attributed to Isaac Goldbring.

At least 37 records · Page 2Linked to original sources
arXiv2024

Undecidability and incompleteness in quantum information theory and operator algebras

We survey a number of incompleteness results in operator algebras stemming from the recent undecidability result in quantum complexity theory known as $\operatorname{MIP}^*=\operatorname{RE}$, the most prominent of which is the Gödelian refutation of the Connes Embedding Problem. We also discuss the very recent use of $\operatorname{MIP}^*=\operatorname{RE}$ in refuting the Aldous-Lyons conjecture in probability theory.

math.LO↗
arXiv2024

Back-and-forth equivalent group von Neumann algebras

We prove that if $G$ and $H$ are $α$-back-and-forth equivalent groups (in the sense of computable structure theory) for some ordinal $α\geq ω$, then their group von Neumann algebras $L(G)$ and $L(H)$ are also $α$-back-and-forth equivalent. In particular, if $G$ and $H$ are $ω$-back-and-forth-equivalent groups, then $L(G)$ and $L(H)$ are elementarily equivalent; this is known to fail under the weaker hypothesis that $G$ and $H$ are merely elementarily equivalent. We extend this result to crossed product von Neumann algebras associated to Bernoulli actions of back-and-forth equivalent groups.

math.LO↗
arXiv2024

Computable Gelfand Duality

We establish a computable version of Gelfand Duality. Under this computable duality, computably compact presentations of metrizable spaces uniformly effectively correspond to computable presentations of unital commutative $C^*$ algebras.

math.LO↗
arXiv2023

A non-uniformly inner amenable group

We provide an example of two elementarily equivalent countable ICC groups $G$ and $H$ such that $G$ is amenable and $H$ is not inner amenable. As a result, we provide the first example of elementarily equivalent groups whose group von Neumann algebras are not elementarily equivalent, answering a question asked by many researchers.

math.GR↗
arXiv2023

The undecidability of having the QWEP

We show that neither the class of C*-algebras with Kirchberg's QWEP property nor the class of W*-probability spaces with the QWEP property are effectively axiomatizable (in the appropriate languages). The latter result follows from a more general result, namely that the hyperfinite III$_1$ factor does not have a computable universal theory in the language of W*-probability spaces. We also prove that the Powers' factors $\mathcal{R}_λ$, for $0<λ<1$, when equipped with their canonical Powers' states, do not have computable universal theory.

math.OA↗
arXiv2023

A Nonstandard Formulation of Bohmian Mechanics

Using the tools of nonstandard analysis, we develop and present an alternative formulation of Bohmian mechanics. This approach allows one to describe a broader assortment of physical systems than the standard formulation of the theory. It also allows one to make predictions in more situations. We motivate the nonstandard formulation with a Bohmian example system that exhibits behavior akin to Earman's (1986) classical space invaders and reverse space invaders. We then use the example to illustrate how the alternative formulation of Bohmian mechanics works.

quant-ph↗
arXiv2023

Uniformly Super McDuff II$_1$ Factors

We introduce and study the family of uniformly super McDuff II$_1$ factors. This family is shown to be closed under elementary equivalence and also coincides with the family of II$_1$ factors with the Brown property introduced in arXiv:2004.02293. We show that a certain family of existentially closed factors, the so-called infinitely generic factors, are uniformly super McDuff, thereby improving a recent result of arXiv:2205.07442. We also show that Popa's family of strongly McDuff II$_1$ factors are uniformly super McDuff. Lastly, we investigate when finitely generic II$_1$ factors are uniformly super McDuff.

math.OA↗
arXiv2023

Locally universal C*-algebras with computable presentations

The Kirchberg Embedding Problem (KEP) asks if every C*-algebra embeds into an ultrapower of the Cuntz algebra $\mathcal{O}_2$. In an effort to provide a negative solution to the KEP and motivated by the recent refutation of the Connes Embedding Problem, we establish two computability-theoretic consequences of a positive solution to KEP. Both of our results follow from the a priori weaker assumption that there exists a locally universal C*-algebra with a computable presentation.

math.LO↗
arXiv2023

A quantum complexity approach to the Kirchberg Embedding Problem

The Kirchberg Embedding Problem (KEP) asks if every C*-algebra embeds into an ultrapower of the Cuntz algebra $\cal O_2$. Motivated by the recent refutation of the Connes Embedding Problem using the quantum complexity result MIP*=RE, we establish two quantum complexity consequences of a positive solution to KEP. Both results involve almost-commuting strategies to nonlocal games.

math.OA↗
arXiv2022

On Tsirelson pairs of C*-algebras

We introduce the notion of a Tsirelson pair of C*-algebras, which is a pair of C*-algebras for which the space of quantum strategies obtained by using states on the minimal tensor product of the pair and the space of quantum strategies obtained by using states on the maximal tensor product of the pair coincide. We exhibit a number of examples of such pairs that are "nontrivial" in the sense that the minimal tensor product and the maximal tensor product of the pair are not isomorphic. For example, we prove that any pair containing a C*-algebra with Kirchberg's QWEP property is a Tsirelson pair. We then introduce the notion of a C*-algebra with the Tsirelson property (TP) and establish a number of closure properties for this class. We also show that the class of C*-algebras with the TP form an axiomatizable class (in the sense of model theory), but that this class admits no "effective" axiomatization.

math.OA↗
arXiv2022

A survey on the model theory of tracial von Neumann algebras

We survey the developments in the model theory of tracial von Neumann algebras that have taken place in the last fifteen years. We discuss the appropriate first-order language for axiomatizing this class as well as the subclass of II$_1$ factors. We discuss how model-theoretic ideas were used to settle a variety of questions around isomorphism of ultrapowers of tracial von Neumann algebras with respect to different ultrafilters before moving on to more model-theoretic concerns, such as theories of II$_1$ factors and existentially closed II$_1$ factors. We conclude with two recent applications of model-theoretic ideas to questions around relative commutants.

math.LO↗
arXiv2022

Approximate traces on groups and the quantum complexity class $\operatorname{MIP}^{co,s}$

An open question in quantum complexity theory is whether or not the class $\operatorname{MIP}^{co}$, consisting of languages that can be efficiently verified using interacting provers sharing quantum resources according to the quantum commuting model, coincides with the class $coRE$ of languages with recursively enumerable complement. We introduce the notion of a qc-modulus, which encodes approximations to quantum commuting correlations, and show that the existence of a computable qc-modulus gives a negative answer to a natural variant of the aforementioned question.

cs.CC↗
arXiv2022

Subsets of Virtually Nilpotent Groups with the SBM Property

We extend Leth's notion of subsets of the integers satisfying the Standard interval measure (SIM) property to the class of virtually nilpotent groups and name the corresponding property the Standard ball measure (SBM) property. In order to do this, we define a natural measure on closed balls in asymptotic cones associated to such groups and show that this measure satisfies the Lebesgue density theorem. We then prove analogs of various properties known to hold for SIM sets in this broader context, occasionally assuming extra properties of the group, such as the small spheres property and the small gaps property.

math.CO↗
arXiv2022

Existentially closed W*-probability spaces

We study several model-theoretic aspects of W$^*$-probability spaces, that is, $σ$-finite von Neumann algebras equipped with a faithful normal state. We first study the existentially closed W$^*$-spaces and prove several structural results about such spaces, including that they are type III$_1$ factors that tensorially absorb the Araki-Woods factor $R_\infty$. We also study the existentially closed objects in the restricted class of W$^*$-probability spaces with Kirchberg's QWEP property, proving that $R_\infty$ itself is such an existentially closed space in this class. Our results about existentially closed probability spaces imply that the class of type III$_1$ factors forms a $\forall_2$-axiomatizable class. We show that for $λ\in (0,1)$, the class of III$_λ$ factors is not $\forall_2$-axiomatizable but is $\forall_3$-axiomatizable; this latter result uses a version of Keisler's Sandwich theorem adapted to continuous logic. Finally, we discuss some results around elementary equivalence of III$_λ$ factors. Using a result of Boutonnet, Chifan, and Ioana, we show that, for any $λ\in (0,1)$, there is a family of pairwise non-elementarily equivalent III$_λ$ factors of size continuum. While we cannot prove the same result for III$_1$ factors, we show that there are at least three pairwise non-elementarily equivalent III$_1$ factors by showing that the class of full factors is preserved under elementary equivalence.

math.OA↗
arXiv2022

The almost sure theory of finite metric spaces

We establish an approximate zero-one law for sentences of continuous logic over finite metric spaces of diameter at most $1$. More precisely, we axiomatize a complete metric theory $T_{\mathrm{as}}$ such that, given any sentence $σ$ in the language of pure metric spaces and any $ε>0$, the probability that the difference of the value of $σ$ in a random metric space of size $n$ and the value of $σ$ in any model of $T_{\mathrm{as}}$ is less than $ε$ approaches $1$ as $n$ approaches infinity. We also establish some model-theoretic properties of the theory $T_{\mathrm{as}}$.

math.LO↗
arXiv2021

Generic algebraic properties in spaces of enumerated groups

We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well known examples from combinatorial group theory, combined with the Baire category theorem, we obtain a plethora of results demonstrating that several phenomena in group theory are generic. In effect, we provide a new topological framework for the analysis of various well known problems in group theory. We also provide a connection between genericity in these spaces, the word problem for finitely generated groups and model-theoretic forcing. Using these connections, we investigate the natural question: when does a certain space of enumerated groups contain a comeager isomorphism class? We obtain a sufficient condition that allows us to answer the question in the negative for the space of all enumerated groups and the space of left orderable enumerated groups. We document several open questions in connection with these considerations.

math.GR↗
arXiv2021

Unitary representations of locally compact groups as metric structures

For a locally compact group $G$, we show that it is possible to present the class of continuous unitary representations of $G$ as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how non-degenerate $*$-representations of a general $*$-algebra $A$ (with some mild assumptions) can be viewed as an elementary class, in a many-sorted language, and use the correspondence between continuous unitary representations of $G$ and non-degenerate $*$-representations of $L^1(G)$. We relate the notion of ultraproduct of logical structures, under this presentation, with other notions of ultraproduct of representations appearing in the literature, and characterise property (T) for $G$ in terms of the definability of the sets of fixed points of $L^1$ functions on $G$.

math.CA↗